simplify the given expression 8x^2-8/ x / 8(x^2 + 8)/26x^2 - 31x

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is 8.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,4 ,8

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 9.00

-2 1 -2.00 12.00

-4 1 -4.00 24.00

-8 1 -8.00 72.00

1 1 1.00 9.00

2 1 2.00 12.00

4 1 4.00 24.00

8 1 8.00 72.00

Polynomial Roots Calculator found no rational roots

Equation at the end of step 1 :

8 (x2+8)

((8•(x2))-((— ÷ 8•——————)•x2))-31x

x 26

:

8

Simplify —

x

Equation at the end of step 2 :

8 (x2+8)

((8•(x2))-((— ÷ 8•——————)•x2))-31x

x 26

8

Divide — by 8

x

1 (x2+8)

((8•(x2))-((—•——————)•x2))-31x

x 26

Equation at the end of step 4 :

(x2 + 8)

((8 • (x2)) - (———————— • x2)) - 31x

26x

Dividing exponential expressions :

x2 divided by x1 = x(2 - 1) = x1 = x

Equation at the end of step 5 :

x • (x2 + 8)

((8 • (x2)) - ————————————) - 31x

26

Equation at the end of step 6 :

x • (x2 + 8)

(23x2 - ————————————) - 31x

26

Rewriting the whole as an Equivalent Fraction :

Subtracting a fraction from a whole

Rewrite the whole as a fraction using 26 as the denominator :

23x2 23x2 • 26

23x2 = ———— = —————————

1 26

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

23x2 • 26 - (x • (x2+8)) -x3 + 208x2 - 8x

———————————————————————— = ————————————————

26 26

Equation at the end of step 7 :

(-x3 + 208x2 - 8x)

—————————————————— - 31x

26

Rewriting the whole as an Equivalent Fraction :

Subtracting a whole from a fraction

Rewrite the whole as a fraction using 26 as the denominator :

31x 31x • 26

31x = ——— = ————————

1 26

Pulling out like terms :

Pull out like factors :

-x3 + 208x2 - 8x = -x • (x2 - 208x + 8)

Trying to factor by splitting the middle term

Factoring x2 - 208x + 8

The first term is, x2 its coefficient is 1 .

The middle term is, -208x its coefficient is -208 .

The last term, "the constant", is +8

Multiply the coefficient of the first term by the constant 1 • 8 = 8

Find two factors of 8 whose sum equals the coefficient of the middle term, which is -208 .

-8 + -1 = -9

-4 + -2 = -6

-2 + -4 = -6

-1 + -8 = -9

1 + 8 = 9

2 + 4 = 6

4 + 2 = 6

8 + 1 = 9

Adding fractions that have a common denominator : Adding up the two equivalent fractions

-x • (x2-208x+8) - (31x • 26) -x3 + 208x2 - 814x

————————————————————————————— = ——————————————————

26 26

Pulling out like terms :

10.1 Pull out like factors :

-x3 + 208x2 - 814x = -x • (x2 - 208x + 814)

Trying to factor by splitting the middle term

10.2 Factoring x2 - 208x + 814

The first term is, x2 its coefficient is 1 .

The middle term is, -208x its coefficient is -208 .

The last term, "the constant", is +814

Multiply the coefficient of the first term by the constant 1 • 814 = 814

Find two factors of 814 whose sum equals the coefficient of the middle term, which is -208 .

-814 + -1 = -815

-407 + -2 = -409

-74 + -11 = -85

-37 + -22 = -59

-22 + -37 = -59

-11 + -74 = -85

-2 + -407 = -409

-1 + -814 = -815

1 + 814 = 815

2 + 407 = 409

11 + 74 = 85

22 + 37 = 59

37 + 22 = 59

74 + 11 = 85

407 + 2 = 409

814 + 1 = 815